Return of the two envelope paradox

[Vraith]

Heh...the real problem with the original problem is that there is no paradox, it isn't a math problem.
The math doesn't change at all...it's 50/50 before you choose, it's 50/50 when you choose one, it's 50/50 if you switch, or if you don't.
The apparent paradox/strategy is a psychology/values problem.
The only reason you CARE if you have .5x, x, or 2x isn't the number, it's cuz it's DOLLARS, and you like/can use more dollars.
If the "prize" was a doctors bill you owe...you'd choose exactly the same strategy for precisely the opposite reason. Cuz you dislike DEBT.

edited for typo that made nonsense instead of mere annoyance.

[TheFallen]

WF...

First of all - you make a mistake when you say the odds of all four are equal at 25%. Because the odds of being in the 3X game vs the 6X game are completely undefined in your problem!

But let's say someone we don't know flips a coin and decides it. Problem solved.

VERY fair point - and that was what I intended when laying out the original apparent paradox, but didn't state. Thanks for the coinflip addition.

However, you have eliminated two possibilities in your original question. You presume you have X, and the other envelop is either 2X or 0.5X.

That's accidental and unintended on my part. I didn't *presume* that I had X - I just decided to call whatever I had in my original envelope X. But to do so and then to further reason as I did is fallacious - as I now understand. And more importantly, I now understand where the fallacy lies.

Thanks for your help.

Hashi...

You are welcome to think so but I disagree with your assessment. My degree in mathematics disagrees with your assessment, as well.

OOOOH an appeal to (self) authority!

You're still point blank wrong - establishing what is in any one single envelope at game start makes absolutely no difference - but I don't have the language to explain any more fully or ably why you are.

I'll leave it to others more mathematically adept to do so - if they can be bothered.

[wayfriend]

As I pointed out above, the culprit is not that you are establishing what is in the envelope. It is that you are a priori selecting what you want the other envelope to contain!

To wit: you are selecting that it must be the small envelope in a big game, or a big envelope in a small game. It must not be the small envelope in the small game, or the big envelope in the big game.

By steering the problem through pre-selection, you alter the outcome. You incur a bias.

I know it doesn't feel like you're pre-selecting. But you are. Math can bite you in the ass on rare occasions. Usually when we are sure we understand things so well that we don't examine them closely.

Vraith: you touch upon what I have called the net of the result vs the odds of the result.

[Hashi Lebwohl]

OOOOH an appeal to (self) authority!

I can think of no higher authority to which to appeal than me.

I'll leave it to others more mathematically adept to do so - if they can be bothered.

I love you, too.

Seriously, though--I think you are overthinking the problem. Given the nature of the problem--your envelope contains x, the other envelope could contain either .5x or 2x--it ultimately doesn't really matter whether you switch or not. The Monty Hall logic says "switch" so you could just go with that.

Your dislike of a conclusion, though, does not mean that the conclusion is incorrect.

[TheFallen]

To wit: you are selecting that it must be the small envelope in a big game, or a big envelope in a small game. It must not be the small envelope in the small game, or the big envelope in the big game.

By steering the problem through pre-selection, you alter the outcome. You incur a bias.

I know it doesn't feel like you're pre-selecting. But you are. Math can bite you in the ass on rare occasions. Usually when we are sure we understand things so well that we don't examine them closely.

Yes WF, that is the heart of my inadvertent fallacy. I've unwittingly excluded two of the possibilities from the set of all possibles.

Now do me another favour please... and tell Hashi that he's wrong in his assertion that opening any single envelope makes the slightest difference to things. He won't take it from me (and I don't blame him... however he is STILL entirely wrong).

I'll leave it to others more mathematically adept to do so - if they can be bothered.

I love you, too.

Heh. I meant others more mathematically adept than ME (and that's a defined set containing a seriously large number).

You're still just plain wrong though. Opening either of the two envelopes at start to establish the value of its contents changes absolutely nothing. It certainly does not transmogrify this seeming paradox into a Monty Hall problem...

[Hashi Lebwohl]

At the very least, the next time someone accuses one of us of blindly agreeing with the other please reference them to this thread. We don't always agree.

[Zarathustra]

I think you all have come to the right conclusion, though you may not be expressing yourselves well. V is wrong, for instance. It is a mathematical problem, because the problem is showing where the logic went wrong, and it went wrong precisely by an incorrect mathematical formulation of the probabilities involved. It takes a correct formulation of the probabilities to correct the logical flaw.

A correct calculation would be:

Expected value in B = 1/2 ( (Expected value in B, given A is larger than B) + (Expected value in B, given A is smaller than B) )

If we then take the sum in one envelope to be x and the sum in the other to be 2x the expected value calculations becomes:

Expected value in B = 1/2 (x + 2x)
which is equal to the expected sum in A.

In non-technical language, what goes wrong (see Necktie paradox) is that, in the scenario provided, the mathematics use relative values of A and B (that is, it assumes that one would gain more money if A is less than B than one would lose if the opposite were true). However, the two values of money are fixed (one envelope contains, say, $20 and the other $40). If the values of the envelopes are restated as x and 2x, it's much easier to see that, if A were greater, one would lose x by switching and, if B were greater, one would gain x by switching. One does not actually gain a greater amount of money by switching because the total T of A and B (3x) remains the same, and the difference x is fixed to T/3.

Two envelopes problem, Simple resolution

[Vraith]

V is wrong, for instance. It is a mathematical problem,

The math is easy, and not paradoxical...pick one, and do what you want, the odds are the same. Which I said, and was correct about. [even if your guy goes into formal explanation].

It SEEMS hard, the "strategy" seems complicated/paradoxical for value/psychological reasons.

[TheFallen]

Vraith, speak for yourself - the maths isn't easy. Well, okay it is, but it's counterintuitive... maybe not for everyone, but I'd bet for the majority.

The bald, inalienable and extremely persuasively misleading fact is that you absolutely know that the envelope opposite you holds either half or double what your own envelope holds. "Easy maths" - or at least too hasty problem framing - leads to an error of thinking that switching is a worthwhile strategy.

Case in point. Hashi still believes that the mere act of opening your own envelope and establishing exactly what it contains, before deciding on whether switching is an advantageous strategy or not, makes a difference to things... and he has a degree in maths.

Ergo, the maths clearly cannot be that simple.

[Vraith]

Vraith, speak for yourself - the maths isn't easy. Well, okay it is, but it's counterintuitive... maybe not for everyone, but I'd bet for the majority.

The bald, inalienable and extremely persuasively misleading fact is that you absolutely know that the envelope opposite you holds either half or double what your own envelope holds. "Easy maths" - or at least too hasty problem framing - leads to an error of thinking that switching is a worthwhile strategy.

Case in point. Hashi still believes that the mere act of opening your own envelope and establishing exactly what it contains, before deciding on whether switching is an advantageous strategy or not, makes a difference to things... and he has a degree in maths.

Ergo, the maths clearly cannot be that simple.

Almost ALL math...and ESPECIALLY statistical---or even stat-esque in appearance--- is counterintuitive. But our intuitiveness is psychological/experiential/anectodal for most people most of the time. The difference between mathematicians and people who can learn to do math [although there is some continua in that] is kinda [don't take that to literally] what they trust...THEN they have to prove/explain it.

your second paragraph is almost exactly what I've said. [[and I didn't intuit this crap...I STRUGGLED with it, till it suddenly started to make sense]]

In SOME situations [the problem here is APPLICATION. WHEN to apply what you know to WHICH situations] H's thoughts in re return on investment and others is correct. But not in THIS situation. When doing this kind of crap minor actual circumstances change things---you can, in a way, know TOO MUCH...you apply what YOU know, but the math/reality/situation is independent of that.

Like this: what are the odds of heads/tails? 1/2 each.
what are the odds of 2 [of either] in a row? 1/4.
So, if you are betting IN ADVANCE, on whether you will get two in a row, you're in a 1/4 situation.
BUT people ASSUME, if they're told the result of a previous toss was heads, the next toss being heads is 1/4 [because of 2 in a row odds].
BUT it's the PERSON who knows the previous result... the coin doesn't know fuck-all. Too much/mis-applied knowledge. The math is simple as a mud pie...dirt and water, 1/2...it's how you think about it, and if what you're thinking applies in particular to THIS problem that makes it hard.

[TheFallen]

Indeed. The issue is with "misapplication" of what one already knows - as you put it - or "misframing of the problem".

Here's a crucially different but superficially nigh identical set-up of the initial question in which switching (just the once) WOULD obviously be advantageous.

Say two envelopes were clearly marked A and B. An unknown (well, unknown to you, that is) amount of money is placed in A. Either double or half that amount is then placed in B, that being decided by a fair coin toss. You are truthfully told of all this in advance. You are then given the envelope clearly marked A and asked whether you wish to switch or not.

In that instance, you SHOULD switch (just the once), because you CAN legitimately define what you're holding in envelope A as X. You KNOW in advance that it's the fixed amount upon which the contents of B were predicated. So it IS correct in this instance to model an equation where A holds X and B holds X/2 or 2X and to act accordingly.

(And again in the above altered problem, opening either envelope before deciding whether to switch or not still of course changes absolutely nothing as to what strategy you should adopt).

But that's NOT the way the original conundrum was formulated. All you know in the original is that your unmarked envelope contains either X or 2X and NOTHING more... which changes things entirely, EVEN THOUGH it is still utterly true that the envelope opposite you holds exactly double or half what yours does.

Assuming that you DO know more than the conundrum itself states is - as you effectively state - where the error comes rushing in. It seems that having too much prior "learning" is sometimes a disadvantage, simply because of the risk that it may lead to faulty assumption. One's previously acquired expertise may be misapplied.

PS I use CAPS above solely in an attempt to increase clarity for any reader as bamboozled as I certainly was.

PPS Hashi... maths degree or no maths degree, you're still totally wrong. Opening any single envelope prior to deciding whether to switch or not in the scenario as originally presented makes precisely zero difference. Regardless of the number of dollars you see, you STILL ONLY know that the opened envelope contains either X or 2X and STILL HAVE NO IDEA which is the case. It is therefore IMPOSSIBLE to come up with any equation in which either envelope can be correctly defined as being of fixed value X in relation to the other

[Vraith]

PS I use CAPS above solely in an attempt to increase clarity for any reader as bamboozled as I certainly was.

Heh...that doesn't bother me at all, though I know some people are bothered by such.
It's bad form, of course, in most writing to use caps/italics/underlines/bold/etc. etc. etc. with rare exceptions. Cuz somebody decided that, and English classes at every level get all persnickity about it....
But I do it here [and similar places] all the time...it's my attempt to sub in tone-of-voice/points of emphasis to the purely printed matter...a kinda/sorta visual analog of speaking style/patterns. It's not REALLY accurate or likely very effective. Use caps the most...which probably misleads, but it's the easiest to do, that convenient key just sitting there. Once upon a time, I reworked my keyboards so along with shift I had Italics key, bold key, undeline key, some other things.

[Zarathustra]

If the math is easy, then show us your math.

Logical fallacies are only psychological in the sense that they occur in someone's mind. They are not psychological in the same sense that bipolar disease is psychological. This is a logic/math problem. It's not a psychological problem. It is solved by correctly describing the probabilities with a mathematical formula, not by correcting a state of mind.

[Vraith]

If the math is easy, then show us your math.

I already did. At least twice.
The chances are 50/50 on your first pick, and they remain 50/50 no matter how many times you switch, from none to infinity. Period.

No, they aren't psychological in the same way as a chemical imbalance or brain damage. I never said that. I very clearly said something entirely different.
I DID describe, loosely, the psych habit/assumptions that I believe makes this easy-peasy thing a "problem."

Try convincing a flat-earther the earth isn't flat with every, any, and all the dozens of "correct formulations"...from multiple and independent methods, lines of evidence and proof...the ONLY way you will do it is by correcting their state of mind.

[[all of that assuming a person who has the mental capacity and desire/interest to understand it, just haven't so far]]

[wayfriend]

So I thought of a sorta kinda succinct way to explain the issue.

There's more money to win when you have the smaller envelope than you stand to lose if you had the larger envelope because there would be twice the amount of money in the game.

[TheFallen]

Try convincing a flat-earther the earth isn't flat with every, any, and all the dozens of "correct formulations"...from multiple and independent methods, lines of evidence and proof...the ONLY way you will do it is by correcting their state of mind.

[[all of that assuming a person who has the mental capacity and desire/interest to understand it, just haven't so far]]

Hashi, I *think* by implication you just got compared to a flat-earther, who needs his state of mind correcting...

WF, that is a succinct way of putting things, but I don't think it's any less confusing... and trust me, I am (or at least was) your poster child for confusion here.

What nailed it for me was:-

a. the two separate "universe" paradigm.

and

b. the inescapable fact being pointed out that you can in fact only have X or 2X... and never know which.

leading to the penny dropping that

c. you can't legitimately describe an utterly indeterminate variable as X and then work it into an equation in the manner that I originally did.

[Vraith]

Hashi, I *think* by implication you just got compared to a flat-earther, who needs his state of mind correcting...

WF, that is a succinct way of putting things, but I don't think it's any less confusing... and trust me, I am (or at least was) your poster child for confusion here.

On the first...that cracked my ass the fuck up. It wasn't at all my intention, I was thinking of other things... but that take is hysterical.

On the second...I think he means the relative "value" difference makes it seem like there is a superior strategy. [my psych thing roughly, I think]...but the correct math is like your b and c.
The defined relationship of the two values isn't relevant.
If you said one envelope has x, the other has z, no one would argue the odds/statistics/strategy. The assigned relationship/value is a red herring.

But the 'net is FULL of people still arguing about it in the most difficult ways possible. At least your penny dropped.
[[I think it only clicked for real for me between last time this issue came up here and this time...I THINK I might have been arguing differently the first time. Though that might have been same problem, different arguing site.]]

[Zarathustra]

V, "show your math" means "show the mathematical reasoning by which you got your answer," not merely restating your answer. As I suspected, you haven't done any math here at all. You just took a guess that happens to be right. No wonder you don't think it's a mathematical issue! I'm not even sure you understand the problem.

You can remove psychology from this completely by restating the problem entirely in terms of maximizing numbers on pieces of paper, rather than talking about money. The same logical flaws would apply.

[Lazy Luke]

Whichever way you cut it, its a 2/1 deal. And whatever math you wish to call it the banker always wins. Its a rigged game!

Here's what you do. Choose an envelope, then burn it. Swap the ashes for the second envelope.
Put the second envelope in your pocket where it will remain, unopened.

Done deal!

[TheFallen]

I've been having fun with this one over the weekend - and causing lots of arguments amongst friends and family.

I did (eventually) find a way to get the penny to drop with the most stubborn (my older brother as it happens). Here's how I did it.

Imagine that there are two pairs of boxes, one red, one green. The red pair we willl call A & B. The green pair we will call C & D.

Red box A is filled with 10 Oreos and red box B is filled with 20 Oreos. Green box C is filled with 20 Twinkies and green box D is filled with 40 Twinkies.

A coin is fairly flipped to decide which pair of boxes (red or green) will be the game offered to the player and a further coin is fairly flipped to see which of the boxes (whose colour has already just been decided) will be given to the player, with its colour-matched pair then being placed opposite the player.

The player of course knows none of this. All he has been told - and thus all he can possibly know - is that his box holds a number of unspecified cookies and that the box opposite him holds either exactly double or exactly half that amount of unspecified cookies.

This then becomes MUCH easier to visualise... well it did for my brother. Because if he started in an Oreo game, it would be IMPOSSIBLE for him to end up with 40 cookies, no matter what he did. Exactly similarly, if he started in a Twinkie game, it would be IMPOSSIBLE for him to end up with 10 cookies, no matter what he did. If you start with Oreos, you CANNOT end up with Twinkies, and vice versa.

Ergo, the fallacy is using X to describe both 20 Oreos AND 20 Twinkies. Regardless of the fact that you absolutely know that the box opposite you contains either precisely double or precisely half the amount of cookies of the box in front of you... you CANNOT then come up with a valid equation that calls the number of cookies in your box "X" and thus the number of cookies in the other box either "X/2" or "2X".

I know everyone here gets this - well, *almost* everyone - so the above exemplar is unnecessary for the Loresraat's purposes. I just thought it was an interesting simplistic take on explanation on where the faulty logic lies. It's a more blunt force way of laying out WF's "two universe" paradigm.

And - just for our resident flat-earther - given the very limited amount that is known by the player, his opening a box to reveal either Oreos or Twinkies AND counting them before deciding if it's worthwhile to switch or not doesn't change a single damn thing.